L'ubomír Snoha

Entropy and periodic points for transitive maps

The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the n–star and the circle among the one–dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.

Minimal Sets on Graphs and Dendrites

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

Topological entropy of Devaney chaotic maps

Abstract The infimum respectively minimum of the topological entropies in different spaces are studied for maps which are transitive or chaotic in the sense of Devaney (i.e., transitive with dense periodic points). After a short survey of results explicitly or implicitly known in the literature for zero and one-dimensional spaces the paper deals with chaotic maps in some higher-dimensional spaces. The key role is played by the result saying that a chaotic map f in a compact metric space X without isolated points can always be extended to a triangular (skew product) map F in X×[0,1] in such a way that F is also chaotic and has the same topological entropy as f. Moreover, the sets X×{0} and X×{1} are F-invariant which enables to use the factorization and obtain in such a way dynamical systems in the cone and in the suspension over X or in the space X× S 1 . This has several consequences. Among others, the best lower bounds for the topological entropy of chaotic maps on disks, tori and spheres of any dimensions are proved to be zero.

Functional envelope of a dynamical system

If (X, f) is a dynamical system given by a compact metric space X and a continuous map f : X → X then by the functional envelope of (X, f) we mean the dynamical system (S(X), Ff) whose phase space S(X) is the space of all continuous selfmaps of X and the map Ff : S(X) → S(X) is defined by Ff() = f for any S(X). The functional envelope of a system always contains a copy of the original system.Our motivation for the study of dynamics in functional envelopes comes from semigroup theory, from the theory of functional difference equations and from dynamical systems theory. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope. Special attention is paid to orbit closures, ω-limit sets, (non)existence of dense orbits and topological entropy.

ON MINIMALITY OF NONAUTONOMOUS DYNAMICAL SYSTEMS

The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a sequence of continuous self-maps of X is studied. A sufficient condition for the nonminimality of such a system is formulated. Special attention is given to the particular case where X is a real compact interval I. A sequence of continuous self-maps of I forming a minimal nonautonomous system may converge uniformly. For example, the limit may be any topologically transitive map. However, if all maps in the sequence are surjective, then the limit is necessarily monotone. An example where the limit is the identity is given. As an application, in a simple way we construct a triangular map in the square I2 with the property that every point except those in the leftmost fiber has an orbit whose ω-limit set coincides with the leftmost fiber.

Almost totally disconnected minimal systems

A space X is said to be almost totally disconnected if the set of its degenerate components is dense in X . We prove that an almost totally disconnected compact metric space admits a minimal map if and only if either it is a finite set or it has no isolated point. As a consequence we obtain a characterization of minimal sets on dendrites and local dendrites.

ON TOPOLOGICAL DYNAMICS OF SEQUENCES OF CONTINUOUS MAPS

We define and study ω-limit sets and topological entropy for a nonautonomous discrete dynamical system given by a sequence

Full cascades of simple periodic orbits on the interval

\{f_i\}_{i =1}^\infty

Special issue dedicated to Francisco Balibrea on the occasion of his 60th birthday

of continuous selfmaps of a compact topological space. A special attention is paid to the case when the space is metric and the sequence

Topological entropy of piecewise bimonotone skew products

\{f_i\}_{i =1}^\infty